In This Article

  1. Introduction
  2. Types of Optimization
  3. Linear Programming
  4. Decision Analysis
  5. Business Applications
  6. Conclusion

Introduction

Optimization is the process of finding the best solution from all feasible solutions. In business, optimization techniques help managers make decisions that maximize profits, minimize costs, or achieve other objectives subject to constraints.

Modern optimization uses mathematical models and computer algorithms to analyze complex problems and find optimal or near-optimal solutions.

Types of Optimization Problems

TypeCharacteristicsExample
Linear ProgrammingLinear objective and constraintsProduction planning
Integer ProgrammingVariables must be integersFacility location
Nonlinear ProgrammingNonlinear functionsPortfolio optimization
Dynamic ProgrammingSequential decisionsInventory management
Stochastic ProgrammingUncertainty in parametersSupply chain planning

Linear Programming

Components of an LP Model

  • Decision Variables: What we're trying to determine
  • Objective Function: What we want to maximize or minimize
  • Constraints: Limitations on the decision variables
  • Non-negativity: Variables cannot be negative

Standard LP Form:

Maximize (or Minimize): Z = c₁x₁ + c₂x₂ + ... + cₙxₙ

Subject to:

a₁₁x₁ + a₁₂x₂ + ... ≤ b₁

a₂₁x₁ + a₂₂x₂ + ... ≤ b₂

x₁, x₂, ... ≥ 0

Example: Product Mix Problem

A company makes products A and B with limited resources:

Maximize: Profit = 30A + 20B

Subject to:

  • 2A + B ≤ 100 (labor hours)
  • A + B ≤ 80 (machine hours)
  • A, B ≥ 0

Solving Methods

  • Graphical method (2 variables)
  • Simplex algorithm
  • Excel Solver

Decision Analysis

Decision Making Under Uncertainty

CriterionApproachRisk Attitude
MaximaxBest of best outcomesRisk-seeking
MaximinBest of worst outcomesRisk-averse
Expected ValueProbability-weighted averageRisk-neutral
Minimax RegretMinimize maximum regretRisk-averse

Decision Trees

Visual tools for sequential decision-making:

  • Decision nodes: Points where you make a choice
  • Chance nodes: Points where uncertainty is resolved
  • Terminal nodes: Final outcomes with payoffs
  • Solve by backward induction (fold back)

Business Applications

  • Production Planning: Optimal product mix and scheduling
  • Supply Chain: Transportation, inventory, facility location
  • Marketing: Budget allocation, media planning
  • Finance: Portfolio optimization, capital budgeting
  • Operations: Workforce scheduling, routing

Conclusion

Key Takeaways

  • Optimization finds best solutions subject to constraints
  • Linear Programming is fundamental for resource allocation
  • LP has objective function, constraints, and decision variables
  • Decision analysis handles uncertainty through probabilities
  • Decision trees help visualize sequential decisions
  • Modern tools like Excel Solver make optimization accessible
  • Applications span production, supply chain, marketing, and finance